Predicting What Comes Next: Sequences and Progressions | EOT

Question 7

The number of bacteria in a certain culture doubles every hour. If there were 30 bacteria present in the culture originally, how many bacteria will be present at the end of the $2^{\text{nd}}$ hour, $4^{\text{th}}$ hour and $n^{\text{th}}$ hour?

Solution

Understand the Question

The culture starts with $30$ bacteria and doubles every hour.
After each hour, the number of bacteria is multiplied by $2$ $2$ :
- After $1$ hour: $30 \times 2^1 = 60$
- After $2$ hours: $30 \times 2^2 = 120$
Following this geometric growth pattern, after $n$ hours, the population will be $30 \times 2^n$ .

(i) Find the number of bacteria present at the end of the $2^{\text{nd}}$ hour.

Step 1 · Calculate Bacteria at the $2^{\text{nd}}$ Hour

Given initial bacteria count $= 30$

Since the number of bacteria doubles every hour

\begin{aligned} \text{Bacteria after 2 hours} &= 30 \times 2^2 \\[0.6em] &= 30 \times 4 \\[0.6em] &= 120 \end{aligned}

Answer

(i) $120$

(ii) Find the number of bacteria present at the end of the $4^{\text{th}}$ hour.

Step 1 · Calculate Bacteria at the $4^{\text{th}}$ Hour

Using the same doubling pattern

\begin{aligned} \text{Bacteria after 4 hours} &= 30 \times 2^4 \\[0.6em] &= 30 \times 16 \\[0.6em] &= 480 \end{aligned}

Answer

(ii) $480$

(iii) Find the number of bacteria present at the end of the $n^{\text{th}}$ hour.

Step 1 · Find Formula for the $n^{\text{th}}$ Hour

After $n$ hours, the initial count of $30$ is multiplied by $2$ a total of $n$ times $\text{Bacteria after } n \text{ hours} = 30 \times 2^n$

Answer

(iii) $30 \times 2^n$

Common Mistakes

Order of Operations (PEMDAS): Do not multiply before evaluating the exponent — $30 \times 2^n \neq (30 \times 2)^n = 60^n$ .
Off-by-One Exponent Error: Since the initial population is at time $t = 0$ , the bacteria count at the end of $n$ hours is $30 \times 2^n$ , not $30 \times 2^{n-1}$ .

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